Definitional property of the integral: for any function
there is a countable sequence of simple functions less than
whose integrals converge to the integral of . (This theorem is
for the most part unnecessary in lieu of itg2i1fseq19528, but unlike that
theorem this one doesn't require to be measurable.) (Contributed
by Mario Carneiro, 14-Aug-2014.)

The integral of a nonnegative constant times a function is the constant
times the integral of the original function. (Contributed by Mario
Carneiro, 28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)

Lemma for itg2mono19526. We show that for any constant less than
one, is less than , and so , which
is one half of the equality in itg2mono19526. Consider the sequence
. This is an increasing
sequence of measurable sets whose union is , and so
has an
integral which equals in the
limit,
by itg1climres19487. Then by taking the limit in
, we get as
desired. (Contributed by Mario Carneiro, 16-Aug-2014.) (Revised by
Mario Carneiro, 23-Aug-2014.)

The Monotone Convergence Theorem for nonnegative functions. If
is a monotone increasing sequence of
positive, measurable, real-valued functions, and is the pointwise
limit of the sequence, then is the limit of the
sequence . (Contributed by
Mario Carneiro, 16-Aug-2014.)

In an extension to the results of itg2i1fseq19528, if there is an upper
bound on the integrals of the simple functions approaching ,
then is real
and the standard limit relation applies.
(Contributed by Mario Carneiro, 17-Aug-2014.)

Special case of itg2i1fseq219529: if the integral of is a real
number, then the standard limit relation holds on the integrals of
simple functions approaching . (Contributed by Mario Carneiro,
17-Aug-2014.)

The integral is
linear. (Measurability is an essential
component of this theorem; otherwise consider the characteristic
function of a nonmeasurable set and its complement.) (Contributed by
Mario Carneiro, 17-Aug-2014.)

The predicate "
is integrable". The "integrable" predicate
corresponds roughly to the range of validity of ,
which is to say that the expression doesn't make sense
unless .
(Contributed by Mario Carneiro,
28-Jun-2014.) (Revised by Mario Carneiro, 23-Aug-2014.)